On a critical Kirchhoff problem in high dimensions

TL;DR

This paper investigates the existence and nonexistence of positive solutions for a high-dimensional Kirchhoff problem involving critical Sobolev exponents, using variational methods to extend and improve previous results.

Contribution

It provides new existence and nonexistence results for the Kirchhoff problem in high dimensions, partially answering an open question for the case N=4 and q>2.

Findings

Established existence of solutions under certain parameter conditions.

Proved nonexistence results for specific parameter regimes.

Extended previous results to higher dimensions and critical cases.

Abstract

In this paper, we consider the following Kirchhoff problem $$ \left\{\aligned -\bigg(a+b\int_{\Omega}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} + \mu u^{2^*-1}, &\quad \text{in }\Omega, \\ u&>0,&\quad\text{in }\Omega,\\ u&=0,&\quad\text{on }\partial\Omega, \endaligned \right.\eqno{(\mathcal{P})} $$ where $\Omega\subset \bbr^N(N\geq4)$ is a bounded domain, $2\leq q<2^*$, $2^*=\frac{2N}{N-2}$ is the critical Sobolev exponent and $a$, $b$, $\lambda$, $\mu$ are positive parameters. By using the variational method, we obtain some existence and nonexistence results to $(\mathcal{P})$ for all $N\geq4$ with some further conditions on the parameters $a$, $b$, $\lambda$, $\mu$, which partially improve some known results in the literatures. Furthermore, Our result for $N=4$ and $q>2$, together with our previous works \cite{HLW15,HLW151}, gives an almost positive answer to Neimen's open question [J. Differential Equations, 257 (2014), 1168--1193].